Flow Matching Policies

Flow matching is a newer way to generate actions, used in recent robot foundation models. Instead of removing noise in many small random steps, it learns a velocity field that carries each noise sample along a nearly straight path to a good action. Following straight paths takes far fewer steps, which matters when a robot must decide many times per second.

Integration steps Samples

A velocity field

Mix noise $\mathbf{x}_0$ and data $\mathbf{x}_1$ along a straight line, $\mathbf{x}_t = (1 - t)\,\mathbf{x}_0 + t\,\mathbf{x}_1$. Train a network $\mathbf{v}_\theta$ to predict the direction of that line:

$$\min_\theta\ \mathbb{E}\,\big\lVert \mathbf{v}_\theta(\mathbf{x}_t, t) - (\mathbf{x}_1 - \mathbf{x}_0)\big\rVert^2 \qquad \frac{d\mathbf{x}}{dt} = \mathbf{v}_\theta(\mathbf{x}, t)$$

To sample, start from noise at $t = 0$ and integrate the field to $t = 1$ with a few Euler steps. Here the field is computed exactly for a known two-mode distribution.

What to notice

Even with very few steps, the samples land close to one of the two valid paths, because the flow from noise to data is nearly straight. Diffusion with the same few steps is still rough. This speed is why some vision-language-action models, such as π0, use a flow matching "action expert" to output smooth action chunks at high rates.